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Κβαντική Ορμή
Ορμή momentum, momentum operator thumb|300px| [[Χαμιλτονιανή Ορμή ]] thumb|300px| [[Φυσική ---- Φυσικοί Γης Νόμοι Φυσικής Νόμοι Φυσικής Θεωρίες Φυσικής Πειράματα Φυσικής Παράδοξα Φυσικής ]] thumb|300px|[[Φυσικό Μέγεθος|Φυσικά Μεγέθη.]] - Ένα Μέγεθος Ετυμολογία Η ονομασία "κβαντική" σχετίζεται ετυμολογικά με την λέξη "Κβάντο". Εισαγωγή In quantum mechanics, momentum is defined as an operator on the wave function. The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once. In quantum mechanics, position and momentum are conjugate variables. For a single particle described in the position basis the momentum operator can be written as : \mathbf{p}={\hbar\over i}\nabla=-i\hbar\nabla\,, :where: * ∇ is the gradient operator, * ħ'' is the reduced Planck constant, and * ''i is the imaginary unit. This is a commonly encountered form of the momentum operator, though the momentum operator in other bases can take other forms. For example, in momentum space the momentum operator is represented as: : \mathbf{p}\psi(p) = p\psi(p)\,, where the operator p''' acting on a wave function ψ''(''p) yields that wave function multiplied by the value p'', in an analogous fashion to the way that the position operator acting on a wave function ''ψ(x'') yields that wave function multiplied by the value ''x. For both massive and massless objects, relativistic momentum is related to the de Broglie wavelength λ'' by : p = h/\lambda.\, Electromagnetic radiation (including visible light, ultraviolet light, and radio waves) is carried by photons. Even though photons (the particle aspect of light) have no mass, they still carry momentum. This leads to applications such as the solar sail. The calculation of the momentum of light within dielectric media is somewhat controversial (see Abraham-Minkowski controversy). Μετασχηματισμός Fourier A three-dimensional wave function in momentum space φ('k) as a weighted sum of orthogonal basis functions φ''j''(k'''): : \phi(\mathbf{k})=\sum_j \psi_j \phi_j(\mathbf{k}) or as an integral: : \phi(\mathbf{k})=\int_{\mathbf{r}{\rm-space}} \psi(\mathbf{r}) \phi_{\mathbf{r}}(\mathbf{k}) {\rm d}^3\mathbf{r} the position operator is given by : \mathbf{\hat r} = i \hbar\frac{\partial}{\partial \bold p} = i\frac{\partial}{\partial \mathbf{k}} with eigenfunctions : \phi_{\mathbf{r}}(\mathbf{k})=\frac{1}{(\sqrt{2\pi})^3} e^{-i \mathbf{k}\cdot\mathbf{r}} and eigenvalues '''r. So a similar decomposition of φ(k') can be made in terms of the eigenfunctions of this operator, which turns out to be the inverse Fourier transform: : \phi(\mathbf{k})=\frac{1}{(\sqrt{2\pi})^3} \int_{\mathbf{r}{\rm-space}} \psi(\mathbf{r}) e^{-i \mathbf{k}\cdot\mathbf{r}} {\rm d}^3\mathbf{r} Τελεστής Ορμής In quantum mechanics, momentum (like all other physical variables) is defined as an operator, which "acts on" or pre-multiplies the wave function to extract the momentum eigenvalue from the wave function: the momentum vector a particle would have when measured in an experiment. The momentum operator is an example of a differential operator. At the time quantum mechanics was developed in the 1920s, the momentum operator was found by many theoretical physicists, including Niels Bohr, Arnold Sommerfeld, Erwin Schrödinger, and Eugene Wigner. Origin from De Broglie plane waves The momentum and energy operators can be constructed in the following way.Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0 One dimension Starting in one dimension, using the plane wave solution to Schrödinger's equation: : \psi = e^{i(kx-\omega t)} The first order partial derivative with respect to space is : \frac{\partial \psi}{\partial x} = i k e^{i(kx-\omega t)} = i k \psi By expressing from the De Broglie relation: : p= \hbar k the formula for the derivative of becomes: : \frac{\partial \psi}{\partial x} = i \frac{p}{\hbar} \psi This suggests the operator equivalence: : \hat{p} = -i\hbar \frac{\partial }{\partial x} so the momentum value is a scalar factor, the momentum of the particle and the value that is measured, is the eigenvalue of the operator. Since the partial derivative is a linear operator, the momentum operator is also linear, and because any wavefunction can be expressed as a superposition of other states, when this momentum operator acts on the entire superimposed wave, it yields the momentum eigenvalues for each plane wave component, the momenta add to the total momentum of the superimposed wave. Three dimensions The derivation in three dimensions is the same, except the gradient operator del is used instead of one partial derivative. In three dimensions, the plane wave solution to Schrödinger's equation is: : \psi = e^{i(\bold{k}\cdot\bold{r}-\omega t)} and the gradient is : \begin{align} \nabla \psi &= \bold{e}_x\frac{\partial \psi}{\partial x} + \bold{e}_y\frac{\partial \psi}{\partial y} + \bold{e}_z\frac{\partial \psi}{\partial z} \\ & = i k_x\psi\bold{e}_x + i k_y\psi\bold{e}_y+ i k_z\psi\bold{e}_z \\ & = \frac{i}{\hbar} \left ( p_x\bold{e}_x + p_y\bold{e}_y+ p_z\bold{e}_z \right)\psi \\ & = \frac{i}{\hbar} \bold{\hat{p}}\psi \end{align} where and are the unit vectors for the three spatial dimensions, hence : \bold{\hat{p}} = -i \hbar \nabla This momentum operator is in position space because the partial derivatives were taken with respect to the spatial variables. Definition (position space) For a single particle with no electric charge and no spin, the momentum operator can be written in the position basis as:''Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10) 0 07 145546 9 : \bold{\hat{p}}=-i\hbar\nabla where is the gradient operator, is the reduced Planck constant, and is the imaginary unit. In one spatial dimension this becomes: : \hat{p}=\hat{p}_x=-i\hbar{\partial \over \partial x}. This is a commonly encountered form of the momentum operator, though not the most general one. For a charged particle in an electromagnetic field, described by the scalar potential and vector potential , the momentum operator must be replaced by:Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0 : \bold{\hat{p}} = -i\hbar\nabla - q\bold{A} where the canonical momentum operator is the above momentum operator: : \bold{\hat{P}} = -i\hbar\nabla This is of course true for electrically neutral particles also, since the second term vanishes if 0}} and the original operator appears. Properties Hermiticity The momentum operator is always a Hermitian operator when it acts on physical (in particular, normalizable) quantum states.See Lecture notes 1 by Robert Littlejohn for a specific mathematical discussion and proof for the case of a single, uncharged, spin-zero particle. See Lecture notes 4 by Robert "Lil' Jon" Littlejohn for the general case. Canonical commutation relation One can easily show that by appropriately using the momentum basis and the position basis: : \left [ \hat{ x }, \hat{ p } \right ] = \hat{x} \hat{p} - \hat{p} \hat{x} = i \hbar. The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once. In quantum mechanics, position and momentum are conjugate variables. Fourier transform One can show that the Fourier transform of the momentum in quantum mechanics is the position operator. The Fourier transform turns the momentum-basis into the position-basis. The following discussion uses the bra–ket notation: : \langle x | \hat{p} | \psi \rangle = - i \hbar \frac{d}{dx} \psi ( x ) The same applies for the position operator in the momentum basis: : \langle p | \hat{x} | \psi \rangle = i \hbar \frac{d}{dp} \psi ( p ) and other useful relations: : \langle p | \hat{x} | p' \rangle = i \hbar \frac{d}{dp} \delta (p - p') : \langle x | \hat{p} | x' \rangle = -i \hbar \frac{d}{dx} \delta (x - x') where stands for Dirac's delta function. Derivation from infinitesimal translations The translation operator is denoted , where represents the length of the translation. It satisfies the following identity: : T(\varepsilon) | \psi \rangle = \int dx T(\varepsilon) | x \rangle \langle x | \psi \rangle that becomes : \int dx | x + \varepsilon \rangle \langle x |\psi \rangle = \int dx | x \rangle \langle x - \varepsilon | \psi \rangle = \int dx | x \rangle \psi(x - \varepsilon) Assuming the function to be analytic (i.e. differentiable in some domain of the complex plane), one may expand in a Taylor series about : : \psi(x-\varepsilon) = \psi(x) - \varepsilon \frac{d \psi}{dx} so for infinitesimal values of : : T(\varepsilon) = 1 - \varepsilon {d \over dx} = 1 - {i \over \hbar} \varepsilon \left ( - i \hbar { d \over dx} \right ) As it is known from classical mechanics, the momentum is the generator of translation, so the relation between translation and momentum operators is: : T(\varepsilon) = 1 - {i \over \hbar} \varepsilon \hat{p} thus : \hat{p} = - i \hbar { d \over dx }. 4-momentum operator Inserting the 3d momentum operator above and the energy operator into the 4-momentum (as a 1-form with metric signature): : P_\mu = \left(\frac{E}{c},-\bold{p}\right) obtains the '''4-momentum operator'; : \hat{P}_\mu = \left(\frac{1}{c}\hat{E},-\bold{\hat{p}}\right) = i\hbar\left(\frac{1}{c}\frac{\partial}{\partial t},\nabla\right) = i\hbar\partial_\mu where is the 4-gradient, and the becomes preceding the 3-momentum operator. This operator occurs in relativistic quantum field theory, such as the Dirac equation and other relativistic wave equations, since energy and momentum combine into the 4-momentum vector above, momentum and energy operators correspond to space and time derivatives, and they need to be first order partial derivatives for Lorentz covariance. The Dirac operator and Dirac slash of the 4-momentum is given by contracting with the gamma matrices: : \gamma^\mu\hat{P}_\mu = i\hbar \gamma^\mu\partial_\mu = \hat{P} = i\hbar\partial If the signature was , the operator would be : \hat{P}_\mu = \left(-\frac{1}{c}\hat{E},\bold{\hat{p}}\right) = -i\hbar\left(\frac{1}{c}\frac{\partial}{\partial t},\nabla\right) = -i\hbar\partial_\mu instead. Υποσημειώσεις Εσωτερική Αρθρογραφία * Παρατηρήσιμο Μέγεθος (observable) * Όργανο Καταμέτρησης * Μονάδα Μέτρησης Βιβλιογραφία * * Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *[ ] *[ ] Κατηγορία:Φυσικά Μεγέθη Κατηγορία:Κβαντική Φυσική